**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXVI, 2 (2007)

p. 179 - 188

Some Comments on Injectivity and p-injectivity

R. Yue Chi Ming

**Abstract**.
A generalization of injective modules (noted GI-modules), distinct from
p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized.
If *M* is a left GI-module, *E* = End (_{A}M), then
*E*/*J*(*E*) is von Neumann regular, where *J*(*E*) is the
Jacobson radical of the ring *E*. *A* is semi-simple Artinian if, and only if, every left *A*-module
is GI. If *A* is a left p. p., left GI-ring such that every non-zero complement left ideal of *A*
contains a non-zero ideal of *A*, then *A* is strongly regular. Sufficient conditions are given for a
ring to be either von Neumann regular or quasi-Frobenius. Quasi-Frobenius and von
Neumann regular rings are characterized. Kasch rings are also considered.

**Keywords:**
injective; GI-module; p-injective; YJ-injective;
von Neumann regular; quasi-Frobenius ring.

**AMS Subject classification.** Primary:16D40, 16D50, 16E50 16D40, 16D50, 16E50.

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